Well yes it will be hard, and hence maybe not economical, and that’s why many people are skeptical of the business case (myself included btw).

But satellite cooling already exists (Starlink v2 satellites dissipate heat at over a kilowatt I believe), so that’s why other people find it plausible.

SpaceX already has 10,000 satellites on orbit that are basically preview versions of space data centers. They've already paid 5 years of that 5-10 year timeline you outlined.

They also need Starship at minimum, which is now a 10+ year old project still exploding regularly.

Starship is at minimum a 2030 project at this point.

And even producing the volume of chips needed for the type of growth space data centers would need to have to justify this would be another decade if construction started now on those fabs.